formation technique to normalize the sector boundaries and congruence transforma- tions in order to re-arrange the matrix inequalities into linear ones. At the end, we obtain a bi-level optimization problem with a high-level problem that is non-convex only in a single scalar variable, while the low-level optimization problem is convex. Hence, we are able to solve the overall problem efficiently using a line search method along with a con- vex optimization method subject to LMI constraints. Hence, the overall problem can be efficiently solved using a line search method along with feasibility checking of LMIs. This is a great advantage over the existing approaches for stability analysis of switched non- linear systems in the literature, which involve searching for Lyapunov functions without a pre-defined structure and/or solving multi-parametric optimization problems [35].
The chapter is organized as follows. In Section 8.2, we present the particular class of switched nonlinear systems under study. Section 8.3 presents stability conditions for the system under arbitrary switching. Section 8.4 discusses stability analysis with an average dwell time constraint. Next, we present the design of robust stabilizing controllers in Section 8.5. We then illustrate the performance of the proposed robust switching control scheme using two examples. Finally, the chapter concludes with a further discussion of the obtained results and open issues.
8.2.PROBLEM
STATEMENT
Consider the following switched nonlinear system: ˙
x(t)=Aσ(t)x(t)+Bσ(t)u(t)+Eσ(t)f¡x(t)¢+Hσ(t)ω(t), (8.1) u(t)=Kσ(t)x(t)+Fσ(t)f¡x(t)¢, (8.2) y(t)=Cσ(t)g¡x(t)¢, (8.3)
withx∈Rn the state vector,u∈Rnu the control input,ω∈Rnωthe disturbance input, y ∈Rny the output, and f :Rn→Rn :xi 7→fi(xi),g:Rn →Rn:xi 7→gi(xi) nonlinear
vector functions. Moreover, the switching signalσis defined as a piecewise constant function,σ(·) : [0,+∞)→{1,... ,N}.
Assumption 8.1. The scalar functions fi are continuous and belong to the classSc1de- fined as follows:
Sc1=©φ:R→R| ∃α,β∈R,α<β, such that¡φ(ζ)−αζ¢¡φ(ζ)−βζ¢≤0,φ(0)=0,∀ζ∈Rª. (8.4) Note that functions fi are not required to lie only in the 1st and the 3rd quadrant as in [82], nor to have unbounded integrals as in [4, 132].
Assumption 8.2. The scalar functions giare continuous and belong to the classSc2de- fined as follows:
Sc2=©ψ:R→R| ∃δsuch that|ψ(ζ)| ≤δ|ζ|,∀ζ∈Rª. (8.5) In fact,Sc2is a special case of the classSc1and functions that belong to the class
Sc2are bounded within a symmetric convex double cone with the origin as apex. More-
8
disturbances. Therefore, specific applications in which this type of disturbances affect the system (e.g. cogging torque or no-current torque, unbalanced gravitational load and eccentricity are state-dependent disturbances affecting a motor control system), can be treated with our proposed analysis and control tools. For the non-switched and simpli- fied version of system (8.1) formulated as follows:
˙
x(t)=E f¡x(t)¢. (8.6) The authors in [132] proved that (8.6) is absolutely stable if there exists a positive definite diagonal matrixΛ=diag{λi},λi>0,∀i∈{1,... ,n}, such that
V(x)= n X i=1 λi Zxi 0 fi(ξ)dξ (8.7)
is a diagonal-type Lyapunov function for (8.6), provided thatxifi(xi)≥0,∀i.
However, stability of a composed switched system cannot be concluded from the stability of the subsystems [149]. According to [149], it is sufficient to construct a com- mon Lyapanov function for a switched system in order to prove stability. Generally, find- ing a common Lyapunov function for the general case of switched nonlinear systems is a tedious task. In [4], stability analysis under arbitrary switching for system (8.1), with
Aℓ=0,∀ℓ∈{1,... ,N}, and withu,ω≡0, using a common Lyapunov function of the form
(8.7) is presented. However, extension of the results for arbitrary switching obtained in [4] to our more general model (8.1)–(8.3) and more important, to the stabilization and robust control problem is not possible. This is mainly because we need to combine and compare the values of the Lyapunov functions and their derivatives in order to compose a stabilizing control law and this is not feasible with the current formulation of the Lya- punov function (8.7) (due to the integral of the nonlinearities). One solution would be to use quadratic functions of the state. However, this choice would increase the conser- vatism in the stability analysis. Therefore, in the following, we use a different Lyapunov function that still contains the nonlinearities in the model and meanwhile, is extend- able for the design of robust stabilizing switching laws. In the first stage, we propose a less conservative approach (compared to the common Lyapunov function method) for stability under arbitrary switching, using the concept of dwell time. Next, we extend the results for state-based switching and design of robust control laws. The resulting design conditions will be formulated in the form of matrix inequalities.
8.3.STABILITYANALYSIS UNDERARBITRARYSWITCHING
For the switched system (8.1) withu(t),ω(t)=0∀t, the following common Lyapunov function is proposed: V(x)=xTP x+2 n X i=1 λi Zxi 0 fi(ξ)dξ. (8.8)
For the Lyapunov functions (8.7) to be radially unbounded, the nonlinear functionsfi
should have an unbounded integral, while in the new Lyapunov function (8.8) this is no longer required and thus, more general cases can be treated through this Lyapunov function. Note that asymptotic stability of all subsystems is a necessary condition for